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Transcript
Examples Example 1 First, find all of the nodes in the circuit. There are four nodes in this circuit. Next, label the nodes. Pick a node to be your reference node. I will pick the node with the most connections. So, V0 will be the reference and I will set its voltage to be equal to 0 V. Label the currents These are the currents that flow through all of the components including the voltage and current sources. Use Kirchhoff’s Current Law Current entering a node must be equal to the current leaving a node. But you only need to use it on nodes V1, V2, and V3. Node V1 I A I B IV Node V2 IV I C 0 A Node V3 I D I I IC Apply Ohm’s Law To rewrite currents in terms of node voltages. The difference in the node voltages across a resistor is equal to its resistance times the current flowing through the resistor. Ohm’s Law does not apply to current or voltage sources. So we can not rewrite IV or II. I A 0V V 1 / 4k V 1 / 4k I B V 1 0V / 2k V 1 / 2k I C V 3 V 2 / 3k I D 0V V 3 / 2k V 3 / 2k Substitute into the KCL equations KCL Equations KCL With Ohm’s Law Node V1: I A I B IV V1 V1 Node V1: IV 4k 2k Node V2 : IV I C 0 A V 3 V 2 Node V2 : IV 0A 3k Node V3: I C I D I I V 3 V 2 V 3 Node V3: II 3k 2k Gather other information from circuit II is equal to 1mA I I 1mA The difference in voltage between V2 and V1 is 5V V 2 V 1 5V or V 2 5V V 1 Substitute this information in 3 equations and three unknowns V1 V1 Node V1: IV 4k 2k V 3 5V V 1 Node V2 : IV 0A 3k V 3 5V V 1 V 3 Node V3: 1mA 3k 2k Solution methods Use MATLAB or other mathematical software package. Use a graphing calculator. Calculate this by hand. You will not have access to MATLAB or other mathematical software or a graphing calculator at the final exam. Make sure that you know how to find the solutions to a set of linear equations by hand. Example 2: Wheatstone Bridge Used in to accurately and precisely determine value of one unknown component. Many times the component’s value changes with an environmental parameter, which allows you to monitor the environmental parameter. For example, a thermister is a resistor whose value is a known function of temperature. If we determine the value of the thermister, we can then calculate the temperature of the thermistor. This technique is used to monitor temperature of furnaces, ovens, refrigerators, and other systems. Vmeas is a function of RL Label nodes and let VD = 0V Note that Vmeas is the difference in voltage between nodes VC and VB: Vmeas = VC - VB Label currents Write Equations KCL: IV I1 I 2 I1 I 3 I2 I L Extra Equations: Ohm’s Law: I1 VA VB / R1 I 2 VA VC / R2 I 3 VB VD / R3 VB / R3 I L VC VD / RL VC / RL Vin VA VD VA Vmeas VC VB Resistor values Usually in Wheatstone Bridge circuits, R1 = R2 = R3 = R Thus, the solution for Vmeas is: RL 1 Vmeas Vin RL R 2 When RL = R, Vmeas = 0V. When RL < R, Vmeas < 0V. When RL > R, Vmeas > 0V. Summary Nodal Analysis is used to calculate the voltage at every node in a circuit and the currents flowing out of or into the voltage sources using a set of simultaneous linear equations. The currents flowing through each resistor can then be calculated using the equations written when applying Ohm’s Law.